P kind, the power acting at a, the weight at c, and the prop or fixed point at B; and because P: W : : CB : AB, P and CB LAB, therefore P= {w, or w = 2P. 191. Corol. 1. Hence it is evident, that when the pul- A B A А B В ley is put in motion, the velocity of the power will be double the velocity of the W weight, as the point p moves twice as fast as the point c and weight w rises. It is also evident, that the fixed pulley x makes no difference in the power p, but is only used to change the direction of it, from upwards to downwards. 192. Corol. 2. Hence we may estimate the effect of a combination of any number of fixed and moveable pulleys ; by which we shall find that every cord going over a moveable pulley always adds 2 to the powers ; since each moveable pulley's rope bears an equal share of the weight; while each rope that is fixed to a pulley, only increases the power by unity. OF THE INCLINED PLANE. 193. THE INCLINED PLANE, is a plane inclined to the horizon, or making an angle with it. It is often reckoned one of the simple mechanic powers ; and the double inclined plane makes the wedge. It is employed to advantage in raising heavy bodies in certain situations, diminishing their weights by laying them on the inclined planes. PROPOSITION or to DW. ABE AB, PROPOSITION XXXV. 194. The Power gained by the Inclined Plane, is in Proportion as the Length of the Plane is to its Height. That is, when Weight w is sustained on an Inclined Plane ; Bo, by a Power P acting in the Direction dw, parallel to the Plane ; then the Weight w, is in proportion to the Power P, as the Length of the Plane is to its Height ; that is, w:P :: BC : AB. Tor, draw AE perp. to DO the plane Bc. Then we are to consider W that the body w is sustained by three forces, viz. Ist, its P A own weight or the force of gravity, acting perp. 10 Ac, or parallel 10 BA ; 2d, by the power p, acting in the direction wd, parallel to BC, or ge; and 3dly, by the re-action of the piane, perp. to its face, or parallel to ihe line Ea. But when a body is kept in equilibrio by the action of three forces, it has been proved, that the intensities of these forces are proportional to the sides of the triangle ABE, made by lines drawn in the directions of their actions ; therefore those forces are to one another as the three lines AB, BE, AE ; that is, the weight of the body w is as the line the power p is as the line BE, and the pressure on the plane as the line As. But the two triangles ABE, ABC are equiangular, and have therefore their like sides proportional ; that is, the three lines AB, BE, AE, are to each other respectively as the three BC, AB, AC, or also as the threc BC, AE, CE, which therefore are as the three forces w, P, , where p denotes the pressure on the plane. That is, w :P :: BC: AB, or the weight is to the power, as the length of the pláne is to its height. See more on the Inclined Plane, at p, 144, &c. 195. Scholium. The Inclined plane comes into use in some situations in which the other mechanical powers cannot be conveniently applied, or in combination with them. As, in sliding heavy weights either up or down a plank or other plane laid sloping : or letting large casks down into a cellar, or drawing them out of it. Also, in removing earth from a lower situation to a higher by means of wheel-barrows, or otherwise, as in making fortifications, &c; inclined planes, made of boards, laid aslope, serve for the barrows to run upon. Of BE Of all the various directions of drawing bodies up an inclined plane, or sustaining them on it, the most favourable is where it is parallel to the plane Bc, and passing through the centre of the weight ; a direction which is easily given to it, by fixing a pulley at D, so that a chord passing over it, and fixed to the weight, may act or draw parallel to the plane. In every other position, it would require a greater power to support the body on the plane, or to draw it up. For if one end of the line be fixed at w, and the other end inclined down towards B, below the direction wd, the body would be drawn down against the plane, and the power must be increased in proportion to the greater difficulty of the traction. And, on the other hand, if the line were carried above the direction of the plane, the power must be also increased ; but here only in proportion as it endeavours to lift the body off the plane. If the length Bc of the plane be equal to any number of times its perp, height ab, as suppose 3 times; then a power p of 1 pound hanging freely, will balance a weight w of 3 pounds, loid on the plane ; and a power P of 2 pounds, will balance a weight w of 6 pounds; and so on, always 3 times as much. But then if they be set a-moving, the perp. descent of the power p, will be equal to 3 times as much as the perp. ascent of the weight w. For, though the weight w ascends up the direction of the oblique plane, bc, just as fast as the power p descends perpendicularly, yet the weight rises only the perp. height AB, while it ascends up the whole length of the plane ec, which is 3 times as much; that is, for every foot of the perp. rise, of the weight, it ascends 3 feet up in the direction of the plane, and the power e descends as much, or 3 feet. PROPOSITION XXXVI. 197. When a Wedge is in Equilibrio ; the Power acting against the Back, is to the Force acting Perfundiculariy against either Side, as the Breadth of the Back AB, is to the Length of the Side ac or BC. For, any three forces, which sustain one A D B another in equilibrio, are as the correspond- E ing sides of a triangle drawn perpendicular to the directions in which they act. But AB is perp. to the force acting on the back, to urge the wedge forward ; and the sides AC, BC are perp. to the forces acting on them ; therefore the three forces are as AB, AC, BC. 198. Corol. The force on the back, AB, which are per.lo them. And therefore the thinner a wedge is, the greater is its effect, in splitting any body, or in overcoming any resistance against the sides of the wedge. DC, 199. But it must be observed, that the resistance, or the forces above-mentioned; respect one side of the wedge only. For if those against both sides be taken in, then, in the foregoing proportions, we must take only half the back id, or else we must take double the line ac or Dc. In the wedge, the friction against the sides is very great, at least equal to the force to be overcome, because the wedge retains any position to which it is driven ; and therefore the resistance is double by the friction. But then the wedge has a great advantage over all the other powers, arising from the force of percussion or blow with which the back is struck, which is a force incomparably greater than any dead weight or pressure, such as is employed in other machines. And accordingly we find it produces effecis vastly superior to those of any other power ; such as the splitting and raising the largest and hardest rocks, the raising and lifting the largest ship, by driving a wedge below it, which a by the blow of a mallet : and thus it appears that the small blow of a hammer, on the back of a wedge, is incomparably greater than any mere pressure, and will overcome it. OF man can do OF THE SCREW. 200, THE Screw is one of the lix mechanical powers, chiefly used in pressing or 'squeezing bodies close, though sometimes also in raising weights. The screw is a spiral thread or groove cut round a cylinder, and every where making the same angle with the length of it. So that if the surface of the cylinder, with this spiral thread on it, were unfolded and stretched into a plane, the spiral thread would form a straight inclined plane, whose length would be to its height, as the circumference of the cylinder, is to the distance between two threads of the screw : as is evident by considering that, in making one round, the spiral rises along the cylinder the distance between the two threads. PROPOSITION XXXVII. 201. The Force of a Power applied to turn a Screw round, is to the Force with which it presses upward or downward, setting aside the Friction, as the Distance between two Threads, is to the Circumference where the Power is applied. The screw being an inclined plane, or half wedge, whose height is the distance between two threads, and its base the circumference of the screw ; and the force in the horizontal direction, being to that in the vertical one, as the lines perpendicular to them, namely, as the height of the plane, or distance of the two threads, is to the base of the plane, or circumference of the screw ; therefore the power is to the pressure, as the distance of two threads is to that circumference. But, by means of a handle or lever, the gain in power is increased in the proportion of the radius of the screw to the radius of the power, or length of the handle, or as their circumferences. Therefore, finally, the power is to the pressure, as the distance of the threads, is to the circumference described by the power. 202. Corol When the screw is put in motion; then the power is to the weight which would keep it in equilibrio, as the velocity of the latter is to that of the former ; and hence their two momenta are equal, which are produced by multiplying each weight or power by its own velocity. So that this is a general property in all the mechanical powers, namely, that the momentum of a power is equal to that of the weight which would balance it in equilibrio ; or that each of them is reciprocally proportional to its velocity. SCHOLIUM |